Single Sample t-test

import polars as pl
import scipy as sp
import numpy as np

# dataset on penguins
# copy from lecture notes: ci example
url = "https://roualdes.sfo3.digitaloceanspaces.com/data/penguins.csv"
df = pl.read_csv(url, null_values = ["NA"]) # this is the trick

bill_length = df.select("bill_length_mm").drop_nulls().to_numpy()

m = np.mean(bill_length)
s = np.std(bill_length)
N = np.size(bill_length)
t = (m - 43) / (s / np.sqrt(N)) # t^*, test-statistic

2 * (1 - sp.stats.t(df = N - 1).cdf(np.abs(t))) # p-value

np.float64(0.0019152230811219084)

Because the p-value < alpha = 0.05, reject H_0. There is evidence that the mean bill length of penguins is not 43mm.

Paired t-test

ndf = (df
 .select("bill_length_mm", "bill_depth_mm")
 .drop_nulls()
 .with_columns((pl.col("bill_length_mm") - pl.col("bill_depth_mm")).alias("d")) # create new cols based on old cols
)

d = ndf["d"].to_numpy()
m = np.mean(d)
s = np.std(d)
N = np.size(d)
t = (m - 0) / (s / np.sqrt(N))
2 * (1 - sp.stats.t(df = N - 1).cdf(np.abs(t))) # p-value

np.float64(0.0)

Based on the data, because the p-value < level of significance, the mean difference (bill_length - bill_depth) is not equal to 0.

t = sp.stats.t(df = N - 1).ppf([0.025, 0.975])
m + t * s / np.sqrt(N)

array([26.10942588, 27.43209458])

Two Sample t-test

sdf = (df
    .select("species", "bill_length_mm")
    .drop_nulls()
)
da = sdf.filter(pl.col("species") == "Adelie")["bill_length_mm"].to_numpy()
dc = sdf.filter(pl.col("species") == "Chinstrap")["bill_length_mm"].to_numpy()

sp.stats.ttest_ind(da, dc)

TtestResult(statistic=np.float64(-23.801939237440887), pvalue=np.float64(2.011759018655462e-62), df=np.float64(217.0))

Because p-value < 0.0001 < alpha = 0.05, we reject H0. There is sufficient evidence to claim that the difference of mean bill length between Adelie and Chinstrap penguins is not equal to zero. On average, Chinstrap and Adelie penguins have differen bill lengths.